OBJ imported into Nifskope has no texture
Not sure what the issue was exactly, but apparently it was with my export. After downgrading to Blender 2.49 and installing the Blender NIF scripts, I was able to export the mesh directly as a NIF (instead of OBJ, then NIF) and now the textures work properly.

By having both versions of Blender installed, I can work in 2.71 with its much-improved interface, then open the *.blend file in 2.49 to export it as a NIF.